dw in AIM Analysis: An Approach to the Nature

Article pubs.acs.org/JPCA

Role of dG/dw and dV/dw in AIM Analysis: An Approach to the Nature of Weak to Strong Interactions Waro Nakanishi* and Satoko Hayashi Department of Material Science and Chemistry, Faculty of Systems Engineering, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan S Supporting Information *

ABSTRACT: Role of dGb(rc)/dw and dVb(rc)/dw is revealed as the basic atoms-inmolecules (AIM) functions to evaluate, classify, and understand the nature of interactions, as well as Gb(rc) and Vb(rc). The border area between van der Waals (vdW) adducts and hydrogen-bonded (HB) adducts is shown to appear at around dGb(rc)/dw = −dVb(rc)/dw and that between molecular complexes (MC) and trigonal bipyramidal adducts (TBP) of chalcogenide dihalides appears at around 2dGb(rc)/dw = −dVb(rc)/dw. Hb(rc) are plotted versus Hb(rc) − Vb(rc)/2 at bond critical points (BCPs) in the AIM dual functional analysis. The plots incorporate the classification of interactions by the signs of ∇2ρb(rc) and Hb(rc). R [= (x2 + y2)1/2] corresponds to the energy for the interaction in question at BCPs, where (x, y) = (Hb(rc), Hb(rc) − Vb(rc)/2) and (x, y) = (0, 0) at the origin. The segment of lines for the plots (S) should correspond to energy, if the segment is substantially linear. The first derivative of S (dS) is demonstrated to be proportional to R. Relations between AIM functions, such as dVb(rc)/dw, dGb(rc)/dw, dHb(rc)/d[Hb(rc) − Vb(rc)/2], d2Vb(rc)/dw2, d2Gb(rc)/dw2, and d2Hb(rc)/d[Hb(rc) − Vb(rc)/2]2, are also discussed. The results help us to understand the nature of interactions.

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INTRODUCTION Atoms-in-molecules method (AIM), proposed by Bader,1,2 enables us to analyze, evaluate, and classify the nature of chemical bonds and interactions.3−7 Electron densities at bond critical points (BCPs: rc, *) of (ω, σ) = (3, −1)1 (ρb(rc)) are strongly related to the binding energies8−14 and bond orders.15 The Laplacian ρb(rc) (∇2ρb(rc)) is the second derivative of ρb(rc); therefore, ρb(rc) is locally depleted relative to the average distribution around rc if ∇2ρb(rc) > 0, but it is concentrated when ∇2ρb(rc) < 0. Total electron energy densities at BCPs (Hb(rc)) must be a more appropriate measure for weak interactions on the energy basis, where Hb(rc) are the sum of kinetic energy densities (Gb(rc)) and potential energy densities (Vb(rc)) at BCPs.1,2,16−20 Electrons at BCPs are stabilized when Hb(rc) < 0; therefore, interactions exhibit the covalent nature in this region, whereas they exhibit no covalency if Hb(rc) > 0 due to destabilization of electrons at BCPs under the conditions. Equations 1 and 2 represent the relations between ∇2ρb(rc) and Hb(rc), together with Gb(rc) and Vb(rc). Hb(rc) must be negative when ∇2ρb(rc) < 0, as confirmed by eq 2 with negative Vb(rc) at all BCPs. (1)

Scheme 1 summarizes the classification of chemical bonds and interactions by the signs of ∇2ρb(rc) and Hb(rc). Interactions in the region of ∇2ρb(rc) < 0 with Hb(rc) < 0 are called shared-shell (SS) interactions, and they are closedshell (CS) interactions for ∇2ρb(rc) > 0. The CS interactions are especially called pure CS interactions for Hb(rc) > 0 and ∇2ρb(rc) > 0. Electrons at BCPs are depleted and destabilized under the conditions.1 Electrons in the intermediate region between SS and pure CS, which belong to CS, are locally depleted but stabilized at BCPs since ∇2ρb(rc) > 0, but Hb(rc) < 0.11 We call the interactions in this region regular CS, when it is necessary to distinguish from pure CS. The role of ∇2ρb(rc) in the classification can be replaced by Hb(rc) − Vb(rc)/2 since signs necessary for the classification are the same with each other as confirmed by eq 2. Scheme 1 also contains the classification by Gb(rc) and Vb(rc) according to eqs 1 and 2. The classification by the signs of ∇2ρb(rc) and Hb(rc) can be achieved only by a parameter θ, as shown in Scheme 1.21 We proposed the AIM dual functional analysis of weak to strong interactions by plotting Hb(rc) versus Hb(rc) − Vb(rc)/ 2,18 after the proposal of Hb(rc) versus ∇2ρb(rc).17 Both treatments are essentially the same with each other and incorporate the classification shown in Scheme 1.19,22 The x and y axes in the plot of Hb(rc) versus Hb(rc) − Vb(rc)/2 are

(2)

Received: September 26, 2012 Revised: January 24, 2013 Published: January 24, 2013

Hb(rc) = G b(rc) + Vb(rc) = (ℏ2 /8m)∇2 ρb (rc) + Vb(rc)/2

(ℏ2 /8m)∇2 ρb (rc) = Hb(rc) − Vb(rc)/2 = G b(rc) + Vb(rc)/2 © 2013 American Chemical Society

1795

dx.doi.org/10.1021/jp3095566 | J. Phys. Chem. A 2013, 117, 1795−1803

The Journal of Physical Chemistry A

Article

Scheme 1. Classification of Interactions by AIM Functions, Together with a Parameter of θ

given in energy unit; therefore, distances on the (x, y) = (Hb(rc) − Vb(rc)/2, Hb(rc)) plane can be expressed in the energy unit. Data for perturbed structures around fully optimized structures are also employed for the plots, together with the fully optimized ones, in our treatment (see Figure 1).18,19,22 The

Figure 1. Polar (R, θ) coordinate representation with (θp, κp) for the plot of H(rc) versus H(rc) − V(rc)/2. Figure 2. Plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 for the vdW, HB, CT−MC, X3−, CT−TBP, and Cov-s interactions, containing Cl3− with a wide range of w represented by a dotted blue line. Numbers are the same as those in Table 1.

perturbed structures are determined to satisfy eq 3, where r and ro are the distances in question for perturbed and fully optimized structures, respectively, with ao of Bohr radius.15 r = ro + wao(w = (0), ±0.05, and ± 0.1; ao = 0.52918 Å) (3)

y/x = (G b(rc) + Vb(rc))/(G b(rc) + Vb(rc)/2)

Figure 1 explains the treatment.18,19,22 Plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 show spiral stream, as a whole (see Figure 2), which are analyzed employing the polar coordinate (R, θ) representation with the (θp, κp) parameters.18,19,22 R in (R, θ), defined by eq 4, corresponds to the energy for an interaction at BCP relative to that without any interaction at the origin in the plot.18,19 θ defined by eq 5 controls the spiral stream of the plot, which is measured from the y-axis. Each plot for an interaction shows a specific curve, which must provide important information about the interaction (see Figure 2). The curve is expressed by (θp, κp): θp corresponds to the tangent line measured from the y-direction as shown by eq 6, and κp is the curvature of the plot as defined by eq 7. The concept of the dynamic nature of interaction is proposed based on the local stream originated from perturbed structures.22 Namely, (R, θ) corresponds to the static nature of interactions, whereas (θp, κp) represents the dynamic nature. Methods to generate the perturbed structures will be discussed later.23 R = (x 2 + y 2 )1/2

(4)

θ = 90° − tan−1(y/x)

(5)

= 2(k + 1)/(k + 2) = 2 − 2/(k + 2)

(5′)

θp = 90° − tan−1(dy/dx)

(6)

κ p = |d2y/dx 2| /[1 + (dy/dx)2 ]3/2

(7)

where (x, y) = (Hb(rc) − Vb(rc)/2, Hb(rc)) The values of Hb(rc) − Vb(rc)/2 and Hb(rc) are necessary to determine the (R, θ) values (eqs 4 and 5), which can be related to Gb(rc) and Vb(rc) according to eqs 1 and 2. If θp are discussed, dHb(rc)/d[Hb(rc) − Vb(rc)/2] are required (eq 6). The values can be derived from dGb(rc)/dw and dVb(rc)/dw, according to eq 8, where dHb(rc)/dw and d[Hb(rc) − Vb(rc)/ 2]/dw are given by eqs 9 and 10, respectively, with w by eq 3. Similarly, (ℏ2/8m)d∇2ρb(rc)/dw can be derived from dGb(rc)/ dw and dVb(rc)/dw, according to eq 11. Second derivatives are contained in eq 7 to evaluate κp, which will be discussed later. The lengths of curved segments (S) in the plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 would have an energy unit, similarly 1796



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Table 1. AIM Functions and Parameters Evaluated for van der Waals Interactions (vdW), Hydrogen Bonds (HB), Molecular Complexes (CT−MC), Trihalide Ions (X3−), Chalcogenide Dihalides of Trigonal Bipyramidal Structures (CT-TBP), Weak Covalent Bonds (Cov-w), and Strong Covalent Bonds (Cov-s) Calculated at the MP2 Levela,b No.

speciesc (X−*−Y)

ro(x, y) (Å)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 No.

He−*−HFl Ne−*−HFl Ar−*−HFl Kr−*−HFl NN−*−HF HF−*−HF HCN−*−HF H2O−*−HOH Me2O−*−HOH Me2O−*−Cl2m Me2O−*−Br2m Me2S−*−Cl2m Me2S−*−Br2m Me2Se−*−Cl2m Me2Se−*−Br2m [Cl−*−Cl2]− [Br−*−Br2]− [Cl−*−BrCl]− [Br−*−ClBr]− Me2ClS−*−Clm Me2BrS−*−Brm Me2ClSe−*−Clm Me2BrSe−*−Brm Me2S+−*−Clm Me2S+−*−Brm Me2Se+−*−Clm Me2Se+−*−Brm Cl−*−Cl Br−*−Br H3C−*−Cl HC−*−CH H2C−*−CH2 H3C−*−CH3 H3C−*−H H−*−H speciesc (X−*−Y)

2.2454 0.0031 2.1982 0.0082 2.5142 0.0066 2.6423 0.0062 2.0293 0.0159 1.8196 0.0252 1.8238 0.0267 1.9427 0.0208 1.8636 0.0263 2.5513 0.0248 2.5913 0.0250 2.6331 0.0273 2.6923 0.0265 2.5700 0.0312 2.7286 0.0257 2.2956 0.0485 2.5474 0.0341 2.4022 0.0424 2.4392 0.0390 2.2650 0.0457 2.4387 0.0354 2.3547 0.0441 2.5196 0.0333 1.9791 0.0716 2.1433 0.0579 2.1089 0.0710 2.2636 0.0486 1.9845 0.0811 2.2690 0.0506 1.7713 0.0715 1.2107 0.2990 1.3349 0.1524 1.5236 0.0661 1.0854 0.0546 0.7366 0.0065 θpp:NIVe (deg) κpp:NIVf

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

l

He−*−HF Ne−*−HFl Ar−*−HFl Kr−*−HFl NN−*−HF HF−*−HF HCN−*−HF H2O−*−HOH Me2O−*−HOH Me2O−*−Cl2 Me2O−*−Br2 Me2S−*−Cl2m Me2S−*−Br2m Me2Se−*−Cl2m Me2Se−*−Br2m [Cl−*−Cl2]− [Br−*−Br2]− [Cl−*−BrCl]− [Br−*−ClBr]− Me2ClS−*−Clm Me2BrS−*−Brm Me2ClSe−*−Clm

57.2 84.3 76.4 80.3 126.3 128.5 167.7 116.7 146.4 98.6 111.7 162.8 171.5 182.5 181.0 180.5 186.5 185.3 183.4 192.8 188.9 186.3

Gb(rc) (au)

Vb(rc) (au)

dGb(rc)/dw (au)

−0.0018 −0.0016 −0.0062 −0.0128 −0.0046 −0.0095 −0.0045 −0.0084 −0.0143 −0.0225 −0.0254 −0.0360 −0.0320 −0.0360 −0.0203 −0.0257 −0.0284 −0.0381 −0.0241 −0.0326 −0.0254 −0.0318 −0.0330 −0.0261 −0.0343 −0.0260 −0.0437 −0.0281 −0.0359 −0.0242 −0.0705 −0.0568 −0.0526 −0.0336 −0.0649 −0.0445 −0.0572 −0.0398 −0.0821 −0.0444 −0.0613 −0.0343 −0.0776 −0.0518 −0.0594 −0.0333 −0.1913 −0.0927 −0.1377 −0.0781 −0.1559 −0.1343 −0.1121 −0.0686 −0.1796 −0.1125 −0.1093 −0.0648 −0.2184 −0.1029 −0.9039 −0.5586 −0.5718 −0.2560 −0.2758 −0.0664 −0.3622 −0.1171 −0.3220 −0.0076 (au−1) Rg (au) θh (deg)

8.66 84.30 163.26 220.67 233.16 103.57 21.99 158.14 88.88 36.05 59.86 49.38 29.49 9.60 13.51 23.53 5.86 4.84 11.00 4.25 3.42 0.23

0.0025 0.0054 0.0048 0.0043 0.0088 0.0125 0.0120 0.0107 0.0123 0.0128 0.0123 0.0122 0.0122 0.0157 0.0129 0.0257 0.0201 0.0246 0.0210 0.0367 0.0262 0.0339

59.9 69.2 65.0 66.5 80.0 90.8 116.1 87.3 99.8 86.8 91.8 117.4 130.1 144.0 142.7 149.0 157.2 156.1 150.1 172.8 169.4 171.0 1797

dVb(rc)/dw (au)

dHb(rc)/dw (au)

0.0008 0.0121 0.0082 0.0076 0.0285 0.0463 0.0610 0.0309 0.0544 0.0349 0.0371 0.0422 0.0460 0.0588 0.0494 0.1147 0.0772 0.0991 0.0849 0.1261 0.0843 0.1183 0.0851 0.3631 0.2324 0.2994 0.1783 0.3409 0.1703 0.4053 1.8821 1.1363 0.4634 0.7946 0.8385 ν (cm−1)

−0.0007 −0.0007 −0.0013 −0.0013 0.0060 0.0102 0.0250 0.0052 0.0164 0.0023 0.0053 0.0161 0.0200 0.0308 0.0252 0.0579 0.0435 0.0546 0.0451 0.0817 0.0500 0.0665 0.0488 0.2704 0.1538 0.1651 0.1097 0.2284 0.1055 0.3024 1.3235 0.8763 0.3970 0.6776 0.8309 kfi (mdyn/Å)

69.1 77.6 70.6 64.0 130.6 166.9 191.5 188.1 176.6 118.0 100.5 104.4 114.9 123.5 108.8 292.5 198.9 248.3 271.5 334.6 358.5 307.8

0.013 0.047 0.039 0.029 0.114 0.081 0.203 0.043 0.052 0.070 0.037 0.044 0.059 0.058 0.078 1.763 1.840 1.721 1.689 0.389 0.294 0.366

dHVb(rc)/dwd (au)

θp:NIVj

dSb(rc)/dw (au)

−0.0012 0.0007 −0.0068 0.0025 −0.0054 0.0016 −0.0046 0.0016 −0.0083 0.0145 −0.0129 0.0251 −0.0055 0.0363 −0.0103 0.0162 −0.0109 0.0347 −0.0152 0.0145 −0.0133 0.0188 −0.0050 0.0364 −0.0030 0.0422 0.0013 0.0610 0.0005 0.0500 0.0006 0.1097 0.0050 0.0822 0.0051 0.1028 0.0027 0.0864 0.0187 0.1475 0.0079 0.0949 0.0074 0.1245 0.0093 0.0924 0.0889 0.4551 0.0381 0.2706 0.0154 0.2938 0.0206 0.1988 0.0580 0.3917 0.0204 0.1918 0.0998 0.5107 0.3825 2.2145 0.3122 1.4555 0.1653 0.6661 0.2802 1.0938 0.4117 1.2939 (deg) κp:NIVk (au−1) sym

57.2 84.3 76.4 80.3 126.3 128.5 167.7 116.7 146.4 98.6 111.7 162.8 171.5 182.3 181.0 180.2 186.5 185.2 183.4 192.8 188.9 186.3

8.13 84.79 161.94 219.23 235.13 103.33 21.60 158.13 89.37 35.54 59.42 51.24 33.00 14.47 13.82 23.75 6.33 5.58 10.74 4.84 2.89 0.57

C∞v C∞v C∞v C∞v C∞v C1i C∞v Cs Cs Cs Cs Cs Cs Cs Cs C∞h C∞h C∞h C∞h C2v C2v C2v



Article

Table 1. continued No. 23 24 25 26 27 28 29 30 31 32 33 34 35

speciesc (X−*−Y)

κpp:NIVf (au−1)

θpp:NIVe (deg)

m

Me2BrSe−*−Br Me2S+−*−Clm Me2S+−*−Brm Me2Se+−*−Clm Me2Se+−*−Brm Cl−*−Cl Br−*−Br H3C−*−Cl HC−*−CH H2C−*−CH2 H3C−*−CH3 H3C−*−H H−*−H

189.0 198.2 193.7 185.3 190.6 194.2 190.9 198.3 196.1 199.4 202.6 202.5 206.4

2.04 0.23 0.21 1.41 0.49 0.63 0.26 0.21 0.01 0.02 0.52 0.04 0.31

Rg (au)

θh (deg)

ν (cm−1)

kfi (mdyn/Å)

θp:NIVj (deg)

0.0264 0.1221 0.0806 0.0852 0.0640 0.0988 0.0588 0.1516 0.6238 0.4402 0.2216 0.3325 0.3512

172.3 191.4 187.8 184.7 186.7 185.0 183.9 194.4 194.2 197.7 198.9 202.4 206.1

233.4 565.4 450.9 465.8 337.8 577.8 342.7 779.0 1968.5 1680.2 1435.0 3203.6 4517.6

0.946 2.456 1.450 4.625 2.952 6.878 5.460 2.509 9.214 4.708 1.540 6.660 12.119

189.0 198.2 193.7 185.6 190.0 194.2 190.9 198.3 195.9 199.4 202.6 202.5 206.4

κp:NIVk (au−1) 1.95 0.16 0.31 1.14 0.37 0.64 0.27 0.19 0.06 0.05 0.54 0.12 0.01

sym C2v Cs Cs Cs Cs C∞h C∞h C3v C∞h C2h D3d Td C∞h

a

The 6-311+G(3df, 3pd) basis set being employed at the MP2 level, unless otherwise noted. bSee also ref 18. cData are given for interaction at BCP, which is shown by * as in NN−*−HF (C∞v). dd[Hb(rc) − Vb(rc)/2]/dw. eCalculated using dGb(rc)/dw and dVb(rc)/dw by applying eqs 6 and 8. f Calculated using dGb(rc)/dw and dVb(rc)/dw according to eq 7, employing eq 17. gDefined by eq 4. hDefined by eq 5. iForce constant corresponding to the frequency. jDefined by eq 6. kDefined by eq 7. lThe pure CS interaction of the vdW type containing the BCP is denoted by −−*−−, while other interaction by −*−. mThe 6-311+G(3d,2p) basis sets being employed only for C and H in CH3. See also ref 22.

to the case of R. Relations between dSb(rc)/dw and R will be examined, which are defined by eqs 12 and 4′, respectively.

frequency analysis performed on the optimized structures. The Møller−Plesset second order energy correlation (MP2) level is applied to the calculations.26,27 Normal coordinates of internal vibrations are employed to generate the perturbed structures. The method is called NIV, which is explained by eq 13. The kth perturbed structures in question (Skw) will be generated by the addition of the normal coordinates of the kth internal vibration (Nk) to the coordinates of the standard orientation of a fully optimized structure (So) in the matrix representation.21,28,29 The coefficient f kw in eq 13 controls the difference in the structures between Skw and So: f kw are determined to satisfy eq 3.30 Nk of five digits are employed to predict Skw.31 AIM functions are calculated with the Gaussian03 program25 and analyzed by the AIM2000 program.32

dHb(rc)/d[Hb(rc) − Vb(rc)/2] = [dHb(rc)/dw]/d[Hb(rc) − Vb(rc)/2]/dw = [dG b(rc)/dw + dVb(rc)/dw] /[dG b(rc)/dw + (1/2)dVb(rc)/dw]

dHb(rc)/dw = dG b(rc)/dw + dVb(rc)/dw

(8) (9)

d[Hb(rc) − Vb(rc)/2]/dw = dG b(rc)/dw + (1/2)dVb(rc)/dw

(10)

(ℏ2 /8m)d∇2 ρb (rc)/dw = dG b(rc)/dw + (1/2)dVb(rc)/dw

Skw = So + fkw Nk

(11)

(13)

y = ao + a1w1 + a 2w 2 + a3w 3 + ... + anw n

dS b(rc)/dw

(Rc 2: square of correlation coefficient)

= [{d(Hb(rc) − Vb(rc)/2)/dw}2 + (dHb(rc)/dw)2 ]1/2 (12) 2

2 1/2

R = {(Hb(rc) − Vb(rc)/2) + Hb(rc) }

(4′)

Here, we discuss and validate the role of dGb(rc)/dw and dVb(rc)/dw in the AIM dual functional analysis, together with Gb(rc) and Vb(rc), for the better understanding of AIM analysis. The results will show that dGb(rc)/dw and dVb(rc)/dw are the key functions in the AIM analysis, together with Gb(rc) and Vb(rc). Conscious efforts have been paid to classify the interactions based on the AIM functions such as ρb(rc), ∇2ρb(rc), Gb(rc), and Vb(rc). However, different kinds of interactions often belong to the same region specified by the functions, if delicate interactions are discussed, such as the case of Mn2(CO)10.24 Applications of dGb(rc)/dw, dVb(rc)/dw, and the related AIM functions will give a hint for the clear classification of interactions.

(14)

dy/dw = a1 + 2a 2w + 3a3w 2 + ... + nanw n − 1

(15)

dy 2 /dw 2 = 2a 2 + 6a3w + ... + nanw n − 1

(16)

d2y/dx 2 = (d/dw)(dw/dx)(dy/dx) = (d2y/dw 2)/(dx /dw)2 − (d2x /dw 2)(dy/dw) /(dx /dw) − (d2x /dw 2)(dy /dw)/(dx /dw)3 (17)

The method to evaluate dGb(rc)/dw and dVb(rc)/dw is as follows: (a) Gb(rc) and Vb(rc) are plotted versus w. (b) Regression curves are determined for the plots with data of the fully optimized structure (w = 0) and the four perturbed ones (w = ± 0.05 and ±0.1), assuming cubic functions as shown in eq 14 (y = Gb(rc) or Vb(rc) with n = 3). (c) The regression curves are differentiated with respect to w, resulting in eqs 15 and 16, where dGb(rc)/dw, dVb(rc)/dw, d2Gb(rc)/dw2, and d2Vb(rc)/dw2 at fully optimized structures are obtained with w = 0. Equation 17 shows the relation for d2y/dx2, which is obtained starting from eqs 15 and 16, where (x, y) = (Hb(rc) −

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METHODOLOGICAL DETAILS IN CALCULATIONS Molecules and adducts are optimized with the 6-311+ +G(3df,3pd) basis sets of the Gaussian03 program,25 unless otherwise noted. The optimized structures are confirmed by the 1798



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Table 2. Requirements for Characteristic Points in the Plots of Hb(rc) versus Hb(rc) − Vb(rc)/2

a

characteristic point

requirements by Hb(rc) and Vb(rc)

requirements by Gb(rc) and Vb(rc)

requirements by ∇2ρb(rc) and Vb(rc)

origin BD1 x-intercept BD2 y-intercept

Hb(rc) − Vb(rc)/2 = Hb(rc) = 0 dHb(rc)/dw = 0 Hb(rc) = 0 2dHb(rc) = Vb(rc)/dw 2Hb(rc) = Vb(rc)/2

Gb(rc) = Vb(rc) = 0 dGb(rc)/dw = −dVb(rc)/dw Gb(rc) = −Vb(rc) 2dGb(rc)/dw = −dVb(rc)/dw 2Gb(rc) = −Vb(rc)

∇2ρb(rc) = Vb(rc) = 0a (ℏ2/4m)d∇2ρb(rc)/dw = −dVb(rc)/dw (ℏ2/4m)∇2ρb(rc) = −Vb(rc) d∇2ρb(rc)/dw = 0 ∇2ρb(rc) = 0a

The same meaning of (ℏ2/8m)∇2ρb(rc) = 0.

Vb(rc)/2, Hb(rc)). The θp values are called θpp:NIV when calculated using dGb(rc)/dw and dVb(rc)/dw by applying eqs 6 and 8, and κp:NIV are named κpp:NIV if calculated using dGb(rc)/ dw and dVb(rc)/dw employing eqs 7 and 17. The partial optimization method (POM) is also applied to generate the perturbed structures, which satisfy eq 3. A wide range of w is employed for r(1Cl−*−2Cl) = ro((1Cl−*−2Cl) + wao for [1Cl−*−2Cl−3Cl]− to show the whole picture of the behavior (see Figure 2). Regression curves of a higher function (n = 9 in eq 14) are used to analyze the wide range of [1Cl−*−2Cl−3Cl]−.

2 = 0, respectively. The requirements are shown by the three forms in Table 2, using dGb(rc)/dw, dVb(rc)/dw, Gb(rc), and Vb(rc), together with ∇2ρb(rc) and Vb(rc). The important role of dGb(rc)/dw and dVb(rc)/dw is well demonstrated in the requirements for the appearance of BD-1 and BD-2, together with that of Gb(rc) and Vb(rc) for the origin, x-intercept, and yintercept. Role of dGb(rc)/dw and dVb(rc)/dw in AIM Analysis. Figure 3 draws the plots of dGb(rc)/dw, −dVb(rc)/dw, Gb(rc), and −V b (r c ) versus w, employing r( 1 Cl−*− 2 Cl) = ro(1Cl−*−2Cl) + wao in [1Cl−*−2Cl−3Cl]−. Data for −dVb(rc)/dw and −Vb(rc) are plotted instead of dVb(rc)/dw and Vb(rc) for convenience of easier understanding of the plots. Monotonic regression curves are for dGb(rc)/dw and −dVb(rc)/ dw versus w, which must be the reflection of the monotonic dependence of Gb(rc) and −Vb(rc) versus w in the region shown in Figure 3. Such behaviors of the functions are desirable for the applications since the high accuracy is required for regression curves. The requirements for the specific points other than the origin are also drawn in Figure 3. Data in Figure 3 approach to the origin in Figure 2 when w becomes very large, where the interactions must be negligibly smaller. How are the requirements in Table 2 visualized in Figure 3, exemplified by 1Cl−*−2Cl? Considerations of the nature for 1 Cl−*−2Cl in Figure 3 should correspond to that of various interactions in Figure 2. The w values at BD-1, x-intercept, BD2, and y-intercept in Figure 3 are 1.52, 0.96, 0.04, and −0.55, respectively, with w = 0.00 for the fully optimized structure. The origin, x-intercept, and y-intercept correspond to Gb(rc) = Vb(rc) = 0, Gb(rc) = Vb(rc), and Gb(rc) = −Vb(rc)/2, respectively. Therefore, 1Cl−*−2Cl in [1Cl−*−2Cl−3Cl]− has the nature of pure CS, regular CS, and SS interactions for the ranges of 0.96 < w, −0.55 < w < 0.96, and w < −0.55, respectively. However, BD-1 and BD-2 appear at w = 1.52 and 0.04, respectively, which correspond to the border area between vdW and HB interactions and that between CT-MC and CTTBP, respectively. Therefore, 1Cl−*−2Cl corresponds to the interaction types of vdW, HB (typical), CT-MC, CT-TBP, and classical covalent bonds of SS for the ranges of 1.52 < w, 0.04 < w < 1.52, 0.04 < w < 0.96, 0.04 < w < −0.55, and w < −0.55, respectively, although some HB are very strong with the SS nature.33 Evaluations of dHb(rc)/d(Hb(rc) − Vb(rc)/2) and d2Hb(rc)/ d(Hb(rc) − Vb(rc)/2)2. As shown in eqs 6 and 7, dy/dx and d2y/ dx2, where (x, y) = (Hb(rc) − Vb(rc)/2, Hb(rc)), are necessary to evaluate the AIM parameters of θp and κp, respectively. They can be obtained directly through regression curves in the plot of dHb(rc) versus d(Hb(rc) − Vb(rc)/2). dHb(rc)/d(Hb(rc) − Vb(rc)/2) can also be derived from the dHb(rc)/dw and d(Hb(rc) − Gb(rc)/2)/dw according to eqs 8 and 10, which are obtained from the plots of Hb(rc) and (Hb(rc) − Gb(rc)/2) versus dw. The process is explained in Figures S2−S4 and eqs S1−S5 in the Supporting Information, exemplified by Br3−. The

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RESULTS AND DISCUSSION AIM Functions and Parameters. Table 1 collects the AIM functions of Gb(rc), Vb(rc), dGb(rc)/dw, dVb(rc)/dw, dHb(rc)/ dw, and d[Hb(rc) − Vb(rc)/2]/dw (abbreviated by dHVb(rc)/ dw in Table 1) with dSb(rc)/dw. Table 1 also contains AIM parameters of (θpp:NIV, κpp:NIV), (R, θ), and (θp:NIV, κp:NIV), together with the optimized interaction distances (ro), frequencies of the internal vibrations (ν) employed to generate the perturbed structures with the force constants (kf), and the symmetry of species.22 Figure 2 draws the plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 mainly for weak interactions using five data of w = 0, ± 0.05, and ±0.1 for each interaction. Figure S1 in the Supporting Information shows the whole picture. Data for wide range of w are also plotted for 1 Cl−*− 2 Cl in [1Cl−*−2Cl−3Cl]− to show the outline of the treatment. How are the plots constructed, which is important since the plots incorporate the classification shown in Scheme 1? No interaction must be detected in the origin in a plot. The plot goes upward due to the increase of Hb(rc) − Vb(rc)/2 and Hb(rc), then arrives at the maximum of Hb(rc). We call this characteristic point the first bending point (BP-1). BP-1 seems to correspond to the border area between the van der Waals (vdW) and hydrogen bonded (HB) interactions. Then the plot bends as Hb(rc) decreases, while Hb(rc) − Vb(rc)/2 increases further, drawing the downward-slope. The plot crosses the xaxis (x-intercepts: Hb(rc) = 0) then reaches the second bending point (BP-2), where Hb(rc) − Vb(rc)/2 is maximum. BP-2 would be the border area between the molecular complex through charge transfer (CT-MC) and trigonal bipyramidal adduct through CT (CT-TBP) interactions in chalcogenide dihalides. After BP-2, the plot goes downward drawing the spiral stream to the right again since both Hb(rc) and Hb(rc) − Vb(rc)/2 decrease. Both Hb(rc) and Hb(rc) − Vb(rc)/2 become negative after the y-intercept where Hb(rc) − Vb(rc)/2 = 0. The final area corresponds to that for the classical chemical bonds of SS. The requirements for the appearance of the specific points in the plots are summarized in Table 2. The requirements for the origin, BD-1, x-intercepts, BD-2, and y-intercepts are Hb(rc) = Hb(rc) − Vb(rc)/2 = 0, dHb(rc)/d[Hb(rc) − Vb(rc)/2] = 0, Hb(rc) = 0, d[Hb(rc) − Vb(rc)/2]/dw = 0, and Hb(rc) − Vb(rc)/ 1799



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dy/dx and d2y/dx2 values are evaluated similarly for some interactions in addition to Br3−, according to eqs 14−17. Table 3 collects the results. The dy/dx and d2y/dx2 values obtained via dGb(rc)/dw and dVb(rc)/dw, through the plots of Gb(rc) and Vb(rc) versus w, are exactly equal to those via the plots of Hb(rc) − Vb(rc)/2 and Hb(rc) versus w. However, d2y/dx2 derived from the direct plots are somewhat different from those obtained above. The differences in d2y/dx2 are around 1% for vdW and HB and 5−10% for CT-MC and CT-TBP. The values are very large for the Cov-w of Me2S+−*−Cl and Cov-s of H3C−*−H, although those for Cov-s of H3C−*−Cl and H3C−*−CH3 are less than 10%. It is noteworthy that the large deviations in d2y/dx2 for the interactions do not affect so much on κp since their κp values are intrinsically very small (see Table 1). What is the reason for the deviations in d2y/dx2? Equation 17 must work well since it is mathematically derived. Some errors in eq 14 (n = 3) would be another reason when it is applied in the analysis. The curvature would be too complex to analyze by eq 14 (n = 3) for some cases, in contrast to the monotonic cases when plotted versus w. If so, the d2y/dx2 values derived from the plots versus w would be more reliable than those obtained directly, although the differences in κp are not so large. The results support well the applicability of dGb(rc)/dw and dVb(rc)/dw for the AIM dual analysis, together with d(Hb(rc) − Vb(rc)/2)/dw and dHb(rc)/dw. AIM analysis by eq 14 (n = 4) with more data would give the better agreement in d2y/dx2. The change in the sign of dy/dx in Table 3 should correspond to the bending points. The change in sign appeared between He−*−HF and NN−*−HF corresponds to BP-1 and that between Me2S−*−Br2 (MC) and [Br−*−Br2]− to BD-2. However, the sign of d2y/dx2 determines the direction of the curvature since the inflection point appears at d2y/dx2 = 0. The curve of the plot should be convex upward for He−*−HF to Me2S−*−Br2, whereas it must be convex downward from [Br−*−Br2]− to H3C−*−Cl, then the curve would be convex upward again for H3C−*−CH3 and H3C−*−H. The results enable us to understand the classification of interactions in more detail. Relations of θpp:NIV versus θp:NIV and κpp:NIV versus κp:NIV. The θpp:NIV values calculated using dGb(rc)/dw and dVb(rc)/dw according to eqs 6 and 8 are plotted versus θp:NIV obtained directly through eq 6. The plot is shown in Figure S5 of the Supporting Information. The plot was analyzed assuming the linear correlation of y = ax + b (a and b being the

Figure 3. Plots of dGb(rc)/dw (●), −dVb(rc)/dw (▲), Gb(rc) (○), and −Vb(rc) (△) versus w in r(Cl−*−Cl) = ro(Cl−*−Cl) + wao for Cl3−: ranges of w are −1.02 ≤ w ≤ 2.10 (a) and 0.70 ≤ w ≤ 1.85 (b).

Table 3. Values of dy/dx and d2y/dx2 Evaluated via the Plots of Gb(rc) and Vb(rc) versus w, x and y versus w, and y versus x, Where (x, y) = (Hb(rc) − Vb(rc)/2, Hb(rc)) via Gb(rc) and Vb(rc) vs w

via x and y vs w

direct method

No.

species

dy/dx

d2y/dx2

dy/dx

d2y/dx2

dy/dx

d2y/dx2

1 5 7 8 13 17 23 24 30 31 34

He−*−HF NN−*−HF HCN−*−HF H2O−*−HOH Me2S−*−Br2 [Br−*−Br2]− Me2BrSe−*−Br Me2S+−*−Cl H3C−*−Cl H3C−*−CH3 H3C−*−H

0.64365 −0.73507 −0.79589 −0.50251 −6.66062 8.80696 6.31928 3.04260 3.03130 2.40153 2.41780

−14.5688 −445.7259 −216.2134 −221.4984 −9010.8804 4078.7486 534.0637 7.4596 6.8360 −9.1999 −0.6600

0.64365 −0.73507 −0.79589 −0.50251 −6.66062 8.80696 6.31928 3.04260 3.03130 2.40153 2.41780

−14.5688 −445.7259 −216.2134 −221.4984 −9010.8804 4078.7486 534.0637 7.4596 6.8360 −9.1999 −0.6600

0.64365 −0.73501 −0.79592 −0.50251 −6.66700 8.82276 6.31683 3.04060 3.03137 2.40162 2.41871

−14.3325 −449.4569 −215.7280 −221.6602 −10111.5208 4432.6540 509.3469 4.7194 6.2646 −9.4644 −2.1161

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correlation constant and y-intercept, respectively, with R2 of the square of correlation coefficient). The correlation was excellent, which is given in eq 18. The magnitudes of the Δθp:NIV (= θpp:NIV − θp:NIV) values are less than 0.6° (see Table 1). The results show that θpp:NIV can also be well evaluated according to eqs 6, 8, and 15, and the values are very close to θp:NIV. Similarly, κpp:NIV evaluated according to eqs 7, 16, and 17 are plotted versus κp:NIV evaluated according to eq 7 employing the dHb(rc)/d(Hb(rc) − Vb(rc)/2) and d2Hb(rc)/d(Hb(rc) − Vb(rc)/2)2 directly obtained through the plot of dHb(rc) versus d(Hb(rc) − Vb(rc)/2). The results are shown in Figure S6 of the Supporting Information. The correlation was also excellent, which is given by eq 19. The results show that κpp:NIV can also be well evaluated through eqs 7, 16, and 17.

to Gb(rc) and Vb(rc). Typical cases of such investigations are in progress.

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S Supporting Information *

Full-optimized structures given by Cartesian coordinates for examined molecules and adducts. This material is available free of charge via the Internet at http://pubs.acs.org.

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*Tel: +81 73 457 8252. Fax: +81 73 457 8253. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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The method using dGb(rc)/dw and dVb(rc)/dw must be superior to the direct one, which is confirmed by the excellent R2 values for Hb(rc) − Vb(rc)/2, Hb(rc), Gb(rc), and Vb(rc) versus w (see eqs S1−S5 of the Supporting Information). The difficulties to evaluate θp at around d(Hb(rc) − Vb(rc)/2) = 0 are also well improved by the method. The relations between AIM functions containing dSb(rc)/dw and R are discussed in the Supporting Information.

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R2 = 0.9999

κ pp:NIV = 1.001κ p:NIV − 0.31

R2 = 0.9997

AUTHOR INFORMATION

Corresponding Author

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θpp:NIV = 1.000θp:NIV − 0.04

ASSOCIATED CONTENT

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ACKNOWLEDGMENTS This work was partially supported by a Grant-in-Aid for Scientific Research (Nos. 20550042, 21550046, and 23350019) from the Ministry of Education, Culture, Sports, Science, and Technology, Japan. The support of the Wakayama University Original Research Support Project Grant and the Wakayama University Graduate School Project Research Grant is also acknowledged. REFERENCES

(1) (a) Bader, R. F. W., Ed. Atoms in Molecules. A Quantum Theory; Oxford University Press: Oxford, U.K., 1990. (b) Matta, C. F.; Boyd, R. J. An Introduction to the Quantum Theory of Atoms in Molecules In The Quantum Theory of Atoms in Molecules: From Solid State to DNA and Drug Design; Matta, C. F., Boyd, R. J., Eds.; Wiley-VCH: Weinheim, Germany, 2007; Chapter 1, p 13. (2) (a) Bader, R. F. W.; Slee, T. S.; Cremer, D.; Kraka, E. Description of Conjugation and Hyperconjugation in Terms of Electron Distributions. J. Am. Chem. Soc. 1983, 105, 5061−5068. (b) Bader, R. F. W. A Quantum Theory of Molecular Structure and Its Applications. Chem. Rev. 1991, 91, 893−926. (c) Bader, R. F. W. A Bond Path: A Universal Indicator of Bonded Interactions. J. Phys. Chem. A 1998, 102, 7314−7323. (d) Biegler-König, F.; Bader, R. F. W.; Tang, T. H. Calculation of the Average Properties of Atoms in Molecules. II. J. Comput. Chem. 1982, 3, 317−328. (e) Bader, R. F. W. Atoms in Molecules. Acc. Chem. Res. 1985, 18, 9−15. (f) Tang, T. H.; Bader, R. F. W.; MacDougall, P. Structure and Bonding in Sulfur− Nitrogen Compounds. Inorg. Chem. 1985, 24, 2047−2053. (g) BieglerKönig, F.; Schönbohm, J.; Bayles, D. Software News and Updates, AIM2000: A Program to Analyze and Visualize Atoms in Molecules. J. Comput. Chem. 2001, 22, 545−559. (h) Biegler-Kö n ig, F.; Schö nbohm, J. Software News and Updates, Update of the AIM2000: Program for Atoms in Molecules. J. Comput. Chem. 2002, 23, 1489−1494. (3) Molina, J.; Dobado, J. A. The Three-Center−Four-Electron (3c− 4e) Bond Nature Revisited. An Atoms-in-Molecules Theory (AIM) and ELF Study. Theor. Chem. Acc. 2001, 105, 328−337. (4) Dobado, J. A.; Martinez-Garcia, H.; Molina, J.; Sundberg, M. R. Chemical Bonding in Hypervalent Molecules Revised. 3. Application of the Atoms in Molecules Theory to Y3X−CH2 (X = N, P, or As; Y = H or F) and H2X−CH2 (X = O, S, or Se) Ylides. J. Am. Chem. Soc. 2000, 122, 1144−1149. (5) Ignatov, S. K.; Rees, N. H.; Tyrrell, B. R.; Dubberley, S. R.; Razuvaev, A. G.; Mountford, P.; Nikonov, G. I. Nonclassical Titanocene Silyl Hydrides. Chem.Eur. J. 2004, 10, 4991−4999. (6) Tripathi, S. K.; Patel, U.; Roy, D.; Sunoj, R. B.; Singh, H. B.; Wolmershäuser, G.; Butcher, R. J. o-Hydroxylmethylphenylchalcogens: Synthesis, Intramolecular Nonbonded Chalcogen···OH Interactions, and Glutathione Peroxidase-Like Activity. J. Org. Chem. 2005, 70, 9237−9247.

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CONCLUSIONS AIM functions of dGb(rc)/dw and dVb(rc)/dw, as well as Gb(rc) and Vb(rc), are demonstrated to play a very important role as basic functions in AIM analysis. Hb(rc) are plotted versus Hb(rc) − Vb(rc)/2 in AIM dual functional analysis, which incorporate the classification of interactions by the signs of ∇2ρb(rc) and Hb(rc) since (ℏ2/8m)∇2ρb(rc) = Hb(rc) − Vb(rc)/ 2. Data of the perturbed structures are also plotted in our treatment, together with those of the fully optimized ones. Interaction distances in question in the perturbed structures (r) must satisfy r = ro + wao (w = ± 0.05 and ±0.1 and ao = 0.52918 Å) with ro in the fully optimized ones. The plots show spiral stream, as a whole, which is characterized by bending points. Requirements for the appearance of the bending points in the plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 can be specified by dGb(rc)/dw and dVb(rc)/dw, while the origin and intercepts in the plots by Gb(rc) and dVb. The functions of dH/d(Hb(rc) − Vb(rc)/2) and d2H/d(Hb(rc) − Vb(rc)/2)2, necessary to analyze the plots, can also be derived from dGb(rc)/dw, dVb(rc)/dw, d2Gb(rc)/d2w, and/or d2Vb(rc)/d2w. The method is shown to be better than the direct one. A linear relationship is confirmed between dSb(rc)/dw and R, although some systematic deviations are observed in weak interactions. The results help us to understand various interactions in more detail. Weak interactions of the HB type often operate to control the subtle conformers. However, it would be difficult to discuss the nature of the interactions, vdW-type or typical HBtype, on the basis of Gb(rc) and Vb(rc) (or Hb(rc) and Hb(rc) − Vb(rc)/2) since such interactions are usually classified only by pure CS. dGb(rc)/dw and dVb(rc)/dw will give a hint to solve the problem, together with the classification of CT-MC from CT-TBP, for example. The nature of transition structures in the reaction processes must be of high interest, which could also be analyzed by applying dGb(rc)/dw and dVb(rc)/dw, in addition 1801



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(24) Bianch, R.; Gervasio, G.; Marabello, D. Experimental Electron Density Analysis of Mn2(CO)10: Metal−Metal and Metal−Ligand Bond Characterization. Inorg. Chem. 2000, 39, 2360−2366. (25) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, revision D.02; Gaussian, Inc.: Wallingford, CT, 2003. (26) Møller, C.; Plesset, M. S. Note on an Approximation Treatment for Many-Electron Systems. Phys. Rev. 1934, 46, 618−622. Gauss, J. Effects of Electron Correlation in the Calculation of Nuclear Magnetic Resonance Chemical Shifts. J. Chem. Phys. 1993, 99, 3629−3643. Gauss, J. Accurate Calculation of NMR Chemical Shifts. Ber. Bunsenges., Phys. Chem. 1995, 99, 1001−1008. (27) Calculations can also be performed at the density functional theory (DFT) level of the Becke three parameter hybrid functionals with the Lee−Yang−Parr correlation functional (B3LYP), if the level is more suitable.35 (28) For the m × n matrix representation, m corresponds to the number of atoms and n (= 3) to the x, y, and z components of the space. (29) The θp and κp values evaluated with NIV are demonstrated to be very close to those corresponding values calculated by POM for the simple interactions such as those shown in Table 1. (30) The values of w = (0), ± 0.1, and ± 0.2 in r = ro + wao were employed for the perturbed structures in POM in refs 10 and 11 since the bond orders becomes 2/3 and 3/2 times larger at w = +0.2 and −0.2 relative to the original values at w = 0, respectively. However, it seems better to employ the perturbed structures as close as possible to the full-optimized ones in NIV. The perturbed structures closer to the full-optimized one will reduce the errors in the AIM functions with the perturbed structures generated by NIV and/or POM. Therefore, w = (0), ± 0.05, and ± 0.1 for r = ro + wao are employed for the analysis in this article. Similarly, w = (0), ± 0.025, and ± 0.05 for θs = θso + wbo are applied to the perturbed structures since ± 0.1bo (±5.73°) would be too large as the perturbations for angles. (31) It is achieved by changing the parameters in Gaussian03 to print out the normal coordinates of five digits for the purpose, although only two digits are printed out as the default. (32) The AIM2000 program (Version 2.0) is employed to analyze and visualize atoms-in-molecules: Biegler-König, F. Calculation of Atomic Integration Data. J. Comput. Chem. 2000, 21, 1040−1048 see also ref 2g. (33) Parthasarathi, R.; Subramanian, V.; Sathyamurthy, N. Hydrogen Bonding without Borders: An Atoms-in-Molecules Perspective. J. Phys. Chem. A 2006, 110, 3349−3351. (34) Nakanishi, W.; Hayashi, S.; Pitak, M. B.; Hursthouse, M. B.; Coles, S. J. Dynamic and Static Behaviors of N−Z−N σ(3c−4e) (Z = S, Se, and Te) Interactions: Atoms-in-Molecules Dual Functional Analysis with High-Resolution X-ray Diffraction Determination of Electron Densities for 2-(2-Pyridylimino)-2H-1,2,4-thiadiazolo[2,3a]pyridine. J. Phys. Chem. A 2011, 115, 11775−11787. (35) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle− Salvetti Correlation-Energy Formula into a Functional of the Electron Density. Phys. Rev. B 1988, 37, 785−789. Miehlich, B.; Savin, A.; Stoll, H.; Preuss, H. Results Obtained with the Correlation Energy Density 1802



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Functionals of Becke and Lee, Yang and Parr. Chem. Phys. Lett. 1989, 157, 200−2006. Becke, A. D. Density-Functional Exchange-Energy Approximation with Correct Asymptotic Behavior. Phys. Rev. A 1988, 38, 3098−3100. Becke, A. D. Density-Functional Thermochemistry. III. The Role of Exact Exchange. J. Chem. Phys. 1993, 98, 5648−5652.

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dw in AIM Analysis: An Approach to the Nature

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